Sufficient conditions for a Dubrovin valuation ring of a simple artinian ring to be nil-clean

Abstract

A subring R of a simple artinian ring Q is said to be a Dubrovin valuation ring if R is a Bezout order and R/J(R) is simple artinian, where J(R) is the Jacobson radical of R. It is known that Q is a simple artinian ring if and only if Q is isomorphic to Mn(D), for some division ring D and n ∈ ℕ. On the other hand, a ring R is said to be nil-clean if each element a ∈ R can be expressed as a = e + n for some idempotent element e ∈ R and nilpotent element n ∈ R. There does not appear to be any studies on connections between these two concepts. In this article, we find sufficient conditions for a Dubrovin valuation ring of a simple artinian ring to be a nil-clean ring.